Depends on what you mean by "Red Giant Sun". By current models of the Sun's evolution, close to its maximum luminosity and radius, the Sun will also be losing mass. In just a few million years it will throw off about 1/4 of its mass, causing the orbits of the planets to increase by ~4/3 fold. However the Earth will be slowed down significantly by tidal forces between it and the bloated Sun, probably causing it to spiral in and be destroyed.

However that's the scenario based on certain mass-loss assumptions. If the Sun sheds a little bit more mass, just a bit quicker, then it might sputter out before it hits that maximum, allowing the Earth to just escape. Of course the Sun sputtering out means it will become a helium white dwarf and slowly chill out over aeons of time. To remain warm enough for life, Earth will need to get a lot closer to the Sun. If the timing is just right, then Earth might be decelerated by the tides just enough to follow the Sun's contraction inwards. Of course during the Red Giant phase, the Sun will be searingly over-bright and Earth is best spiraling away from the Sun for the ~billion year long climb to the Red Giant Tip. Developing the means of shifting planets at will in the next 7 billion years would be very handy.

In just a few million years it will throw off about 1/4 of its mass, causing the orbits of the planets to increase by ~4/3 fold. However the Earth will be slowed down significantly by tidal forces between it and the bloated Sun, probably causing it to spiral in and be destroyed.

Where did you get this infomation, I've heard this theory as well however I would like to know where you found this because it doesn't make sense. No evidence what so ever that it will spiral in.

We revisit the distant future of the Sun and the solar system, based on stellar models computed with a thoroughly tested evolution code. For the solar giant stages, mass-loss by the cool (but not dust-driven) wind is considered in detail. Using the new and well-calibrated mass-loss formula of Schroder & Cuntz (2005, 2007), we find that the mass lost by the Sun as an RGB giant (0.332 M_Sun, 7.59 Gy from now) potentially gives planet Earth a significant orbital expansion, inversely proportional to the remaining solar mass.
According to these solar evolution models, the closest encounter of planet Earth with the solar cool giant photosphere will occur during the tip-RGB phase. During this critical episode, for each time-step of the evolution model, we consider the loss of orbital angular momentum suffered by planet Earth from tidal interaction with the giant Sun, as well as dynamical drag in the lower chromosphere. We find that planet Earth will not be able to escape engulfment, despite the positive effect of solar mass-loss. In order to survive the solar tip-RGB phase, any hypothetical planet would require a present-day minimum orbital radius of about 1.15 AU.
Furthermore, our solar evolution models with detailed mass-loss description predict that the resulting tip-AGB giant will not reach its tip-RGB size. The main reason is the more significant amount of mass lost already in the RGB phase of the Sun. Hence, the tip-AGB luminosity will come short of driving a final, dust-driven superwind, and there will be no regular solar planetary nebula (PN). But a last thermal pulse may produce a circumstellar (CS) shell similar to, but rather smaller than, that of the peculiar PN IC 2149 with an estimated total CS shell mass of just a few hundredths of a solar mass.

Notice their conclusion, that Earth would survive the RGB Tip - which lasts merely ~100,000s of years at its extreme - if just 1.15 AU from the Sun. Given 7.59 billion years to do so and a requisite dv of just 2010 m/s that means a continually force of 50 GN will put the Earth at 1.15 AU (or equivalent) as the Sun hits the RGB Tip. The expected mass loss will mean Earth will end up at ~1.72 AU, thus experiencing an insolation peak of ~921 times the present. If it could be enclosed in a shell with a reflectivity of ~0.999 then it'd still be habitable.

Once the RGB Tip is past, the Sun will hit the Helium Main Sequence and shine at ~55 times present luminosity, with Earth experiencing ~18.56 times its present insolation. A reflectivity of 0.95 will do.

However if the Sun's mass-loss can be tweaked a bit, then it'll end the RGB early and not undergo the Helium Flash transition to the Helium Main Sequence. Instead it'll become a Helium white-dwarf. According to Eric Agol's work on the white dwarf habitable zone, adjusted for a helium white dwarf, that means potentially ~20 billion years before the Sun gets too cold to orbit safely, if Earth can be induced to spiral in sufficiently closely. Riding the tides of the shrinking solar envelope will be quite a challenge, but we have about 7 billion years to learn how.

However long before the sun goes into its red giant phase the oceans will evaporate. The sun warms as it moves through main sequence phase and luminosity increase by about 10% every billion years. So in 2 billion years time the ocean will have already boiled away and life will have become impossible except inside the crust where strangley organisms are found. In 250 million year when (probably) a new supercontinent forms the sun will be warmer. Can you imagine the hurricanes. Life on earth is doomed long before the Sun dies.

However long before the sun goes into its red giant phase the oceans will evaporate. The sun warms as it moves through main sequence phase and luminosity increase by about 10% every billion years. So in 2 billion years time the ocean will have already boiled away and life will have become impossible except inside the crust where strangley organisms are found. In 250 million year when (probably) a new supercontinent forms the sun will be warmer. Can you imagine the hurricanes. Life on earth is doomed long before the Sun dies.

Assumes (a) oceans will remain at present volume and (b) the same atmospheric pressure for all time. Some geophysical models suggest (a) & (b) are perhaps erroneous. With less surface water and less nitrogen partial pressure the Earth will avoid the runaway greenhouse effect. As a 'desert planet' with some water near the Poles it will remain habitable until the Sun starts ascending the Red giant branch. It'll be different to the present day Earth, but still viable for some kinds of life.